Question

Lease term=3

monthly rental payments made in advance

residual payment of 40% of the initial value of the car

P: initial value of the car = $50,000

n:the term in months= 36

i: interest rate on the lease per month=0.75%

X: residual payment= 0.40×$50,000 = $20,000.

R: monthly rentals

Show that this final payment received by the leasing company can be rewritten as X-Max(X-S,0) which can be thought of as the residual payment less the payoff on a put option

Answer 1

At any point, the lease holder would still want to keep making the residual payments, if the residual value of the car (S) is still greater than the residual value of the payments left. i.e. as long as S>X, lease holder will keep making the payments.

So, From the leasing company perspective, Their payment is actually Min(X, S)

We can re-write Min(X,S) as

= X - Max(X-S,0) .............................................. (1)

To illustrate this, lets look at below two scenarios

1. X<S

In this case, as Residual payment is less than residual value of Car, 2nd part of the expression (1) is Max (-ve number, 0) = 0.

i..e Payment received by leasing company = X

2. X>S i.e X-S>0

In this case, 2nd part of the equation will be Max (X-S,0)

so, Payment = X - Max(X-S,0)

=X - (X-S) (since X-S is now >0)

= S

So, final payment received by the leasing company can be rewritten as X-Max(X-S,0)